As you mention, the hypothesis ($\bullet$) implies that for every singular cardinal $\kappa$ and every inner model $W$ of $\mathsf{ZFC}$, we have $\kappa^{+W} < \kappa^+$. From this consequence, I suspect that it might follow by the argument of Busche and Schindler [1] that there is a proper class model of $\mathsf{AD}$ in a forcing extension of $V$.

The reason I say this is that in the paper [1, Subsections 2.2.2 and 2.2.3] it seems like this consequence might suffice for their argument when it is applied to the $\mathsf{ZFC}$ model $W = \text{Lp}(A_0)$ for a sufficiently closed singular cardinal $\kappa$ and a sufficiently complicated set of ordinals $A_0 \subset \kappa$ that they consider. The notation $\text{Lp}(A_0)$ refers to the "lower part mouse" over $A_0$, which is the relevant generalization of $L$ in the argument for $0^\sharp$ that you mention.

I haven't read the paper carefully, so if you want to know whether this is true then you should ask them.

The next natural goal for a consistency strength lower bound would be $\mathsf{AD}_\mathbb{R}$. I don't think anyone knows exactly how to get this from ($\bullet$) at the moment; the absence of choice may limit the available methods somewhat. It might be a good problem. (Again the relevant consequence would probably be that inner models of choice do not compute successors of singulars correctly.)

In terms of large cardinals, the theory $\mathsf{ZF} + \mathsf{AD}$ is equiconsistent with $\mathsf{ZFC} + {}$"there are infinitely many Woodin cardinals," and $\mathsf{ZF} + \mathsf{AD}_\mathbb{R}$ is equiconsistent with $\mathsf{ZFC} + {}$"there is a cardinal $\lambda$ that is a limit of Woodin cardinals and $\mathord{<}\lambda$-strong cardinals." However, current methods typically proceed in terms of building determinacy models rather than building models of $\mathsf{ZFC} + {}$large cardinals directly.

It may be interesting to note that ($\bullet$) implies $\neg\square_\kappa$ for every infinite cardinal $\kappa$ (so in particular we get failures of square at singular cardinals.) The reason is that from a square sequence we can recursively define a sequence of surjections $f_\alpha :\kappa \to \alpha$ for $\alpha < \kappa^+$, using our clubs at limit stages to piece together the surjections that we have already constructed into a new surjection. (Coherence is superfluous.) Then we can define an injection from $\kappa^+$ to $\mathcal{P}(\kappa \times \kappa)$, or equivalently to $\mathcal{P}(\kappa)$, by $\alpha \mapsto \{(\gamma,\delta) \in \kappa \times \kappa : f_\alpha(\gamma) < f_\alpha(\delta)\}$.

[1] Busche, Daniel, and Ralf Schindler. The strength of choiceless patterns of singular and weakly compact cardinals, Annals of Pure and Applied Logic, 159(1), 198–248, 2009